Random dynamical system

In the mathematical field of dynamical systems, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are characterized by a state space S, a set of maps T from S into itself that can be thought of as the set of all possible equations of motion, and a probability distribution Q on the set T that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state $X\in S$ evolving according to a succession of maps randomly chosen according to the distribution Q.^[1]

An example of a random dynamical system is a stochastic differential equation; in this case the distribution Q is typically determined by noise terms. It consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space.

Motivation: solutions to a stochastic differential equation

Let $f:\mathbb {R} ^{d}\to \mathbb {R} ^{d}$ be a $d$ -dimensional vector field, and let $\varepsilon >0$ . Suppose that the solution $X(t,\omega ;x_{0})$ to the stochastic differential equation

\left\{{\begin{matrix}\mathrm {d} X=f(X)\,\mathrm {d} t+\varepsilon \,\mathrm {d} W(t);\\X(0)=x_{0};\end{matrix}}\right.

exists for all positive time and some (small) interval of negative time dependent upon $\omega \in \Omega$ , where $W:\mathbb {R} \times \Omega \to \mathbb {R} ^{d}$ denotes a $d$ -dimensional Wiener process (Brownian motion). Implicitly, this statement uses the classical Wiener probability space

(\Omega ,{\mathcal {F}},\mathbb {P} ):=\left(C_{0}(\mathbb {R} ;\mathbb {R} ^{d}),{\mathcal {B}}(C_{0}(\mathbb {R} ;\mathbb {R} ^{d})),\gamma \right).

In this context, the Wiener process is the coordinate process.

Now define a flow map or (solution operator) $\varphi :\mathbb {R} \times \Omega \times \mathbb {R} ^{d}\to \mathbb {R} ^{d}$ by

\varphi (t,\omega ,x_{0}):=X(t,\omega ;x_{0})

(whenever the right hand side is well-defined). Then $\varphi$ (or, more precisely, the pair $(\mathbb {R} ^{d},\varphi )$ ) is a (local, left-sided) random dynamical system. The process of generating a "flow" from the solution to a stochastic differential equation leads us to study suitably defined "flows" on their own. These "flows" are random dynamical systems.

Formal definition

Formally, a random dynamical system consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. In detail.

Let $(\Omega ,{\mathcal {F}},\mathbb {P} )$ be a probability space, the noise space. Define the base flow $\vartheta :\mathbb {R} \times \Omega \to \Omega$ as follows: for each "time" $s\in \mathbb {R}$ , let $\vartheta _{s}:\Omega \to \Omega$ be a measure-preserving measurable function:

\mathbb {P} (E)=\mathbb {P} (\vartheta _{s}^{-1}(E))

for all

E\in {\mathcal {F}}

and

s\in \mathbb {R}

;

Suppose also that

$\vartheta _{0}=\mathrm {id} _{\Omega }:\Omega \to \Omega$ , the identity function on $\Omega$ ;
for all $s,t\in \mathbb {R}$ , $\vartheta _{s}\circ \vartheta _{t}=\vartheta _{s+t}$ .

That is, $\vartheta _{s}$ , $s\in \mathbb {R}$ , forms a group of measure-preserving transformation of the noise $(\Omega ,{\mathcal {F}},\mathbb {P} )$ . For one-sided random dynamical systems, one would consider only positive indices $s$ ; for discrete-time random dynamical systems, one would consider only integer-valued $s$ ; in these cases, the maps $\vartheta _{s}$ would only form a commutative monoid instead of a group.

While true in most applications, it is not usually part of the formal definition of a random dynamical system to require that the measure-preserving dynamical system $(\Omega ,{\mathcal {F}},\mathbb {P} ,\vartheta )$ is ergodic.

Now let $(X,d)$ be a complete separable metric space, the phase space. Let $\varphi :\mathbb {R} \times \Omega \times X\to X$ be a $({\mathcal {B}}(\mathbb {R} )\otimes {\mathcal {F}}\otimes {\mathcal {B}}(X),{\mathcal {B}}(X))$ -measurable function such that

for all $\omega \in \Omega$ , $\varphi (0,\omega )=\mathrm {id} _{X}:X\to X$ , the identity function on $X$ ;
for (almost) all $\omega \in \Omega$ , $(t,\omega ,x)\mapsto \varphi (t,\omega ,x)$ is continuous in both $t$ and $x$ ;
$\varphi$ satisfies the (crude) cocycle property: for almost all $\omega \in \Omega$ ,

\varphi (t,\vartheta _{s}(\omega ))\circ \varphi (s,\omega )=\varphi (t+s,\omega ).

In the case of random dynamical systems driven by a Wiener process $W:\mathbb {R} \times \Omega \to X$ , the base flow $\vartheta _{s}:\Omega \to \Omega$ would be given by

W(t,\vartheta _{s}(\omega ))=W(t+s,\omega )-W(s,\omega )

This can be read as saying that $\vartheta _{s}$ "starts the noise at time $s$ instead of time 0". Thus, the cocycle property can be read as saying that evolving the initial condition $x_{0}$ with some noise $\omega$ for $s$ seconds and then through $t$ seconds with the same noise (as started from the $s$ seconds mark) gives the same result as evolving $x_{0}$ through $(t+s)$ seconds with that same noise.

Attractors for random dynamical systems

The notion of an attractor for a random dynamical system is not as straightforward to define as in the deterministic case. For technical reasons, it is necessary to "rewind time", as in the definition of a pullback attractor. Moreover, the attractor is dependent upon the realisation $\omega$ of the noise.

References

↑ Bhattacharya, Rabi, and Mukul Majumdar. "Random dynamical systems: a review." Economic Theory 23, no. 1 (2003): 13-38., Rabi; Mukul Majumdar (2003). "Random dynamical systems: a review". Economic Theory. 23 (1): 13–38. doi:10.1007/s00199-003-0357-4. Retrieved 26 June 2013.

Crauel, H., Debussche, A., & Flandoli, F. (1997) Random attractors. Journal of Dynamics and Differential Equations. 9(2) 307—341.
Arnold, L (1998/2003) Random Dynamical Systems. Springer Monographs in Mathematics

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

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Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise

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Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

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Random dynamical system

Motivation: solutions to a stochastic differential equation

Formal definition

Attractors for random dynamical systems

See also

References